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  <svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg></a></div> <!----></nav>  <ul class="sidebar-links"><li><div class="sidebar-group first"><p class="sidebar-heading open"><span>学术型算法题解</span> <!----></p> <ul class="sidebar-group-items"><li><a href="/03.Software/01.DataStructureAndAlgorithm/Others/Part02.%E5%AD%A6%E6%9C%AF%E5%9E%8B%E7%AE%97%E6%B3%95%E9%A2%98%E8%A7%A3.html#目录" class="sidebar-link">目录</a><ul class="sidebar-sub-headers"></ul></li><li><a href="/03.Software/01.DataStructureAndAlgorithm/Others/Part02.%E5%AD%A6%E6%9C%AF%E5%9E%8B%E7%AE%97%E6%B3%95%E9%A2%98%E8%A7%A3.html#_01-蓄水池抽样" class="sidebar-link">01.蓄水池抽样</a><ul class="sidebar-sub-headers"><li class="sidebar-sub-header"><a href="/03.Software/01.DataStructureAndAlgorithm/Others/Part02.%E5%AD%A6%E6%9C%AF%E5%9E%8B%E7%AE%97%E6%B3%95%E9%A2%98%E8%A7%A3.html#_1-1-382-链表随机节点『纯模板』" class="sidebar-link">1.1.382. 链表随机节点『纯模板』</a></li><li class="sidebar-sub-header"><a href="/03.Software/01.DataStructureAndAlgorithm/Others/Part02.%E5%AD%A6%E6%9C%AF%E5%9E%8B%E7%AE%97%E6%B3%95%E9%A2%98%E8%A7%A3.html#_1-2-398-随机数索引" class="sidebar-link">1.2.398. 随机数索引</a></li><li class="sidebar-sub-header"><a href="/03.Software/01.DataStructureAndAlgorithm/Others/Part02.%E5%AD%A6%E6%9C%AF%E5%9E%8B%E7%AE%97%E6%B3%95%E9%A2%98%E8%A7%A3.html#_1-3-519-随机翻转矩阵" class="sidebar-link">1.3.519. 随机翻转矩阵</a></li><li class="sidebar-sub-header"><a href="/03.Software/01.DataStructureAndAlgorithm/Others/Part02.%E5%AD%A6%E6%9C%AF%E5%9E%8B%E7%AE%97%E6%B3%95%E9%A2%98%E8%A7%A3.html#_1-4-497-非重叠矩形中的随机点" class="sidebar-link">1.4.497. 非重叠矩形中的随机点</a></li></ul></li><li><a href="/03.Software/01.DataStructureAndAlgorithm/Others/Part02.%E5%AD%A6%E6%9C%AF%E5%9E%8B%E7%AE%97%E6%B3%95%E9%A2%98%E8%A7%A3.html#_02-拒绝采样" class="sidebar-link">02.拒绝采样</a><ul class="sidebar-sub-headers"><li class="sidebar-sub-header"><a href="/03.Software/01.DataStructureAndAlgorithm/Others/Part02.%E5%AD%A6%E6%9C%AF%E5%9E%8B%E7%AE%97%E6%B3%95%E9%A2%98%E8%A7%A3.html#_2-1-470-用-rand7-实现-rand10" class="sidebar-link">2.1.470. 用 Rand7() 实现 Rand10()</a></li></ul></li><li><a href="/03.Software/01.DataStructureAndAlgorithm/Others/Part02.%E5%AD%A6%E6%9C%AF%E5%9E%8B%E7%AE%97%E6%B3%95%E9%A2%98%E8%A7%A3.html#_03-数据流" class="sidebar-link">03.数据流</a><ul class="sidebar-sub-headers"><li class="sidebar-sub-header"><a href="/03.Software/01.DataStructureAndAlgorithm/Others/Part02.%E5%AD%A6%E6%9C%AF%E5%9E%8B%E7%AE%97%E6%B3%95%E9%A2%98%E8%A7%A3.html#_3-1-1352-最后-k-个数的乘积" class="sidebar-link">3.1.1352. 最后 K 个数的乘积</a></li></ul></li></ul></div></li></ul> </div> <div class="page"> <div class="content"><h1 id="学术型算法题解"><a href="#学术型算法题解" class="header-anchor">#</a> 学术型算法题解</h1> <h2 id="目录"><a href="#目录" class="header-anchor">#</a> 目录</h2> <p>[TOC]</p> <h2 id="_01-蓄水池抽样"><a href="#_01-蓄水池抽样" class="header-anchor">#</a> 01.蓄水池抽样</h2> <ul><li>水塘抽样/<a href="https://leetcode-cn.com/tag/reservoir-sampling/problemset/" target="_blank" rel="noopener noreferrer">蓄水池抽样<svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg></a></li></ul> <div class="language-cpp line-numbers-mode"><pre class="language-cpp"><code>问题描述
如何从一个未知行随机选择一行<span class="token operator">?</span>

隐藏含义<span class="token operator">:</span>
<span class="token number">1</span>、只能顺序读取每一行『我们不能随机的读文件中的某一行』
<span class="token number">2</span>、不能将整个文件加载到内存『因为文件可能特别大』
</code></pre> <div class="line-numbers-wrapper"><span class="line-number">1</span><br><span class="line-number">2</span><br><span class="line-number">3</span><br><span class="line-number">4</span><br><span class="line-number">5</span><br><span class="line-number">6</span><br></div></div><img src="/assets/img/01.蓄水池算法1.4a1f473b.png"> <ul><li>计算概率</li></ul> <img src="/assets/img/01.蓄水池算法2计算.25f543c4.png"> <h3 id="_1-1-382-链表随机节点『纯模板』"><a href="#_1-1-382-链表随机节点『纯模板』" class="header-anchor">#</a> 1.1.<a href="https://leetcode-cn.com/problems/linked-list-random-node/" target="_blank" rel="noopener noreferrer">382. 链表随机节点<svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg></a>『纯模板』</h3> <ul><li><a href="https://leetcode-cn.com/problems/linked-list-random-node/" target="_blank" rel="noopener noreferrer">382. 链表随机节点<svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg></a></li> <li>选取k=1的蓄水池抽样『模板代码』</li></ul> <div class="language-cpp line-numbers-mode"><pre class="language-cpp"><code><span class="token comment">/**
 * Definition for singly-linked list.
 * struct ListNode {
 *     int val;
 *     ListNode *next;
 *     ListNode() : val(0), next(nullptr) {}
 *     ListNode(int x) : val(x), next(nullptr) {}
 *     ListNode(int x, ListNode *next) : val(x), next(next) {}
 * };
 */</span>
<span class="token keyword">class</span> <span class="token class-name">Solution</span> <span class="token punctuation">{</span>
<span class="token keyword">public</span><span class="token operator">:</span>
    <span class="token comment">/** @param head The linked list's head.
        Note that the head is guaranteed to be not null, so it contains at least one node. */</span>
    <span class="token function">Solution</span><span class="token punctuation">(</span>ListNode<span class="token operator">*</span> head<span class="token punctuation">)</span> <span class="token punctuation">{</span>
        inner_head<span class="token operator">=</span>head<span class="token punctuation">;</span>
    <span class="token punctuation">}</span>
    
    <span class="token comment">/** Returns a random node's value. */</span>
    <span class="token keyword">int</span> <span class="token function">getRandom</span><span class="token punctuation">(</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
        <span class="token keyword">int</span> selectRes<span class="token operator">=</span><span class="token number">0x3f3f3f</span><span class="token punctuation">;</span><span class="token comment">//表示不选择</span>
        <span class="token comment">//『易错的地方，一定要用临时变量，不能修改inner_head，我也不知道为啥』</span>
        ListNode <span class="token operator">*</span> loop<span class="token operator">=</span>inner_head<span class="token punctuation">;</span> 
        <span class="token keyword">int</span> curNum<span class="token operator">=</span><span class="token number">1</span><span class="token punctuation">;</span>
        <span class="token comment">// rand() % k == 0 模拟概率为 1/k 的情况</span>
        <span class="token keyword">while</span><span class="token punctuation">(</span> <span class="token keyword">nullptr</span><span class="token operator">!=</span>loop <span class="token punctuation">)</span>
        <span class="token punctuation">{</span>
            <span class="token keyword">if</span><span class="token punctuation">(</span> <span class="token number">0</span><span class="token operator">==</span><span class="token function">rand</span><span class="token punctuation">(</span><span class="token punctuation">)</span><span class="token operator">%</span>curNum <span class="token punctuation">)</span>
            <span class="token punctuation">{</span>
                selectRes<span class="token operator">=</span>loop<span class="token operator">-&gt;</span>val<span class="token punctuation">;</span>
            <span class="token punctuation">}</span>
            <span class="token operator">++</span>curNum<span class="token punctuation">;</span>
            loop<span class="token operator">=</span>loop<span class="token operator">-&gt;</span>next<span class="token punctuation">;</span>
        <span class="token punctuation">}</span>

        <span class="token keyword">return</span> selectRes<span class="token punctuation">;</span>
    <span class="token punctuation">}</span>
<span class="token keyword">private</span><span class="token operator">:</span>
    ListNode <span class="token operator">*</span> inner_head<span class="token punctuation">;</span>
<span class="token punctuation">}</span><span class="token punctuation">;</span>

<span class="token comment">/**
 * Your Solution object will be instantiated and called as such:
 * Solution* obj = new Solution(head);
 * int param_1 = obj-&gt;getRandom();
 */</span>
</code></pre> <div class="line-numbers-wrapper"><span class="line-number">1</span><br><span class="line-number">2</span><br><span class="line-number">3</span><br><span class="line-number">4</span><br><span class="line-number">5</span><br><span class="line-number">6</span><br><span class="line-number">7</span><br><span class="line-number">8</span><br><span class="line-number">9</span><br><span class="line-number">10</span><br><span class="line-number">11</span><br><span class="line-number">12</span><br><span class="line-number">13</span><br><span class="line-number">14</span><br><span class="line-number">15</span><br><span class="line-number">16</span><br><span class="line-number">17</span><br><span class="line-number">18</span><br><span class="line-number">19</span><br><span class="line-number">20</span><br><span class="line-number">21</span><br><span class="line-number">22</span><br><span class="line-number">23</span><br><span class="line-number">24</span><br><span class="line-number">25</span><br><span class="line-number">26</span><br><span class="line-number">27</span><br><span class="line-number">28</span><br><span class="line-number">29</span><br><span class="line-number">30</span><br><span class="line-number">31</span><br><span class="line-number">32</span><br><span class="line-number">33</span><br><span class="line-number">34</span><br><span class="line-number">35</span><br><span class="line-number">36</span><br><span class="line-number">37</span><br><span class="line-number">38</span><br><span class="line-number">39</span><br><span class="line-number">40</span><br><span class="line-number">41</span><br><span class="line-number">42</span><br><span class="line-number">43</span><br><span class="line-number">44</span><br><span class="line-number">45</span><br><span class="line-number">46</span><br></div></div><h3 id="_1-2-398-随机数索引"><a href="#_1-2-398-随机数索引" class="header-anchor">#</a> 1.2.<a href="https://leetcode-cn.com/problems/random-pick-index/" target="_blank" rel="noopener noreferrer">398. 随机数索引<svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg></a></h3> <ul><li><a href="https://leetcode-cn.com/problems/random-pick-index/" target="_blank" rel="noopener noreferrer">398. 随机数索引<svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg></a></li> <li>纯模板的改进</li></ul> <div class="language-cpp line-numbers-mode"><pre class="language-cpp"><code><span class="token keyword">class</span> <span class="token class-name">Solution</span> <span class="token punctuation">{</span>
<span class="token keyword">public</span><span class="token operator">:</span>
    <span class="token function">Solution</span><span class="token punctuation">(</span>vector<span class="token operator">&lt;</span><span class="token keyword">int</span><span class="token operator">&gt;</span><span class="token operator">&amp;</span> nums<span class="token punctuation">)</span> <span class="token operator">:</span><span class="token function">inner_nums</span><span class="token punctuation">(</span>nums<span class="token punctuation">)</span> <span class="token punctuation">{</span>

    <span class="token punctuation">}</span>
    
    <span class="token keyword">int</span> <span class="token function">pick</span><span class="token punctuation">(</span><span class="token keyword">int</span> target<span class="token punctuation">)</span> <span class="token punctuation">{</span>
        <span class="token keyword">int</span> count<span class="token operator">=</span><span class="token number">1</span><span class="token punctuation">;</span>
        <span class="token keyword">int</span> res<span class="token operator">=</span><span class="token number">0</span><span class="token punctuation">;</span>  <span class="token comment">//假装是第0号</span>
        <span class="token keyword">for</span><span class="token punctuation">(</span><span class="token keyword">int</span> loop<span class="token operator">=</span><span class="token number">0</span><span class="token punctuation">;</span> loop<span class="token operator">&lt;</span>inner_nums<span class="token punctuation">.</span><span class="token function">size</span><span class="token punctuation">(</span><span class="token punctuation">)</span><span class="token punctuation">;</span> <span class="token operator">++</span>loop<span class="token punctuation">)</span>
        <span class="token punctuation">{</span>
            <span class="token comment">//和纯模板不同的地方，只管那个我们想要管的</span>
            <span class="token keyword">if</span><span class="token punctuation">(</span> target<span class="token operator">==</span>inner_nums<span class="token punctuation">[</span>loop<span class="token punctuation">]</span> <span class="token punctuation">)</span>
            <span class="token punctuation">{</span>
                <span class="token comment">// rand() % k == 0 模拟概率为 1/k 的情况</span>
                <span class="token keyword">if</span><span class="token punctuation">(</span> <span class="token number">0</span><span class="token operator">==</span><span class="token function">rand</span><span class="token punctuation">(</span><span class="token punctuation">)</span><span class="token operator">%</span>count <span class="token punctuation">)</span>
                <span class="token punctuation">{</span>
                    res<span class="token operator">=</span>loop<span class="token punctuation">;</span>
                <span class="token punctuation">}</span>
                <span class="token operator">++</span>count<span class="token punctuation">;</span>
            <span class="token punctuation">}</span>
        <span class="token punctuation">}</span>

        <span class="token keyword">return</span> res<span class="token punctuation">;</span>
    <span class="token punctuation">}</span>
<span class="token keyword">private</span><span class="token operator">:</span>
    vector<span class="token operator">&lt;</span><span class="token keyword">int</span><span class="token operator">&gt;</span> inner_nums<span class="token punctuation">;</span>
<span class="token punctuation">}</span><span class="token punctuation">;</span>

<span class="token comment">/**
 * Your Solution object will be instantiated and called as such:
 * Solution* obj = new Solution(nums);
 * int param_1 = obj-&gt;pick(target);
 */</span>
</code></pre> <div class="line-numbers-wrapper"><span class="line-number">1</span><br><span class="line-number">2</span><br><span class="line-number">3</span><br><span class="line-number">4</span><br><span class="line-number">5</span><br><span class="line-number">6</span><br><span class="line-number">7</span><br><span class="line-number">8</span><br><span class="line-number">9</span><br><span class="line-number">10</span><br><span class="line-number">11</span><br><span class="line-number">12</span><br><span class="line-number">13</span><br><span class="line-number">14</span><br><span class="line-number">15</span><br><span class="line-number">16</span><br><span class="line-number">17</span><br><span class="line-number">18</span><br><span class="line-number">19</span><br><span class="line-number">20</span><br><span class="line-number">21</span><br><span class="line-number">22</span><br><span class="line-number">23</span><br><span class="line-number">24</span><br><span class="line-number">25</span><br><span class="line-number">26</span><br><span class="line-number">27</span><br><span class="line-number">28</span><br><span class="line-number">29</span><br><span class="line-number">30</span><br><span class="line-number">31</span><br><span class="line-number">32</span><br><span class="line-number">33</span><br><span class="line-number">34</span><br></div></div><h3 id="_1-3-519-随机翻转矩阵"><a href="#_1-3-519-随机翻转矩阵" class="header-anchor">#</a> 1.3.<a href="https://leetcode-cn.com/problems/random-flip-matrix/" target="_blank" rel="noopener noreferrer">519. 随机翻转矩阵<svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg></a></h3> <ul><li><a href="https://leetcode-cn.com/problems/random-flip-matrix/" target="_blank" rel="noopener noreferrer">519. 随机翻转矩阵<svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg></a></li></ul> <h3 id="_1-4-497-非重叠矩形中的随机点"><a href="#_1-4-497-非重叠矩形中的随机点" class="header-anchor">#</a> 1.4.<a href="https://leetcode-cn.com/problems/random-point-in-non-overlapping-rectangles/" target="_blank" rel="noopener noreferrer">497. 非重叠矩形中的随机点<svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg></a></h3> <ul><li><a href="https://leetcode-cn.com/problems/random-point-in-non-overlapping-rectangles/" target="_blank" rel="noopener noreferrer">497. 非重叠矩形中的随机点<svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg></a></li></ul> <h2 id="_02-拒绝采样"><a href="#_02-拒绝采样" class="header-anchor">#</a> 02.拒绝采样</h2> <ul><li><a href="https://leetcode-cn.com/tag/rejection-sampling/problemset/" target="_blank" rel="noopener noreferrer">拒绝采样<svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg></a></li></ul> <div class="language-txt line-numbers-mode"><pre class="language-text"><code>在数值分析和计算统计中，拒绝采样是用于从分布生成观测值的基本技术。
它通常也被称为接受拒绝方法或「接受拒绝算法」，是一种精确的模拟方法。该方法适用于具有密度的任何分布。

拒绝采样基于以下观察：在一维中对随机变量进行采样，可以对二维笛卡尔图执行均匀随机采样，并将样本保持在其密度函数图下的区域中。

拒绝采样是蒙特卡罗方法的一种，在众多的采样方法中，这种方法显得独树一帜。而且，这种方法与Metropolis-Hastings算法也有一定关系
</code></pre> <div class="line-numbers-wrapper"><span class="line-number">1</span><br><span class="line-number">2</span><br><span class="line-number">3</span><br><span class="line-number">4</span><br><span class="line-number">5</span><br><span class="line-number">6</span><br></div></div><h3 id="_2-1-470-用-rand7-实现-rand10"><a href="#_2-1-470-用-rand7-实现-rand10" class="header-anchor">#</a> 2.1.<a href="https://leetcode-cn.com/problems/implement-rand10-using-rand7/" target="_blank" rel="noopener noreferrer">470. 用 Rand7() 实现 Rand10()<svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg></a></h3> <ul><li><a href="https://leetcode-cn.com/problems/implement-rand10-using-rand7/" target="_blank" rel="noopener noreferrer">470. 用 Rand7() 实现 Rand10()<svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg></a></li> <li>『体会到概率论的code实现。。。』</li></ul> <div class="language-cpp line-numbers-mode"><pre class="language-cpp"><code><span class="token comment">// The rand7() API is already defined for you.</span>
<span class="token comment">// int rand7();</span>
<span class="token comment">// @return a random integer in the range 1 to 7</span>

<span class="token keyword">class</span> <span class="token class-name">Solution</span> <span class="token punctuation">{</span>
<span class="token keyword">public</span><span class="token operator">:</span>
    <span class="token keyword">int</span> <span class="token function">rand10</span><span class="token punctuation">(</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
        <span class="token keyword">int</span> row<span class="token punctuation">,</span> col<span class="token punctuation">,</span> idx<span class="token punctuation">;</span>
        <span class="token keyword">do</span> <span class="token punctuation">{</span>
            row <span class="token operator">=</span> <span class="token function">rand7</span><span class="token punctuation">(</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
            col <span class="token operator">=</span> <span class="token function">rand7</span><span class="token punctuation">(</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
            idx <span class="token operator">=</span> col <span class="token operator">+</span> <span class="token punctuation">(</span>row <span class="token operator">-</span> <span class="token number">1</span><span class="token punctuation">)</span> <span class="token operator">*</span> <span class="token number">7</span><span class="token punctuation">;</span>
        <span class="token punctuation">}</span> <span class="token keyword">while</span> <span class="token punctuation">(</span>idx <span class="token operator">&gt;</span> <span class="token number">40</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
        <span class="token keyword">return</span> <span class="token number">1</span> <span class="token operator">+</span> <span class="token punctuation">(</span>idx <span class="token operator">-</span> <span class="token number">1</span><span class="token punctuation">)</span> <span class="token operator">%</span> <span class="token number">10</span><span class="token punctuation">;</span>
    <span class="token punctuation">}</span>
<span class="token punctuation">}</span><span class="token punctuation">;</span>
</code></pre> <div class="line-numbers-wrapper"><span class="line-number">1</span><br><span class="line-number">2</span><br><span class="line-number">3</span><br><span class="line-number">4</span><br><span class="line-number">5</span><br><span class="line-number">6</span><br><span class="line-number">7</span><br><span class="line-number">8</span><br><span class="line-number">9</span><br><span class="line-number">10</span><br><span class="line-number">11</span><br><span class="line-number">12</span><br><span class="line-number">13</span><br><span class="line-number">14</span><br><span class="line-number">15</span><br><span class="line-number">16</span><br></div></div><h2 id="_03-数据流"><a href="#_03-数据流" class="header-anchor">#</a> 03.数据流</h2> <ul><li><a href="https://leetcode-cn.com/tag/data-stream/problemset/" target="_blank" rel="noopener noreferrer">数据流<svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg></a></li></ul> <p>表示文件可能特别大，无法存入内存。</p> <ul><li><a href="https://leetcode-cn.com/problems/shu-ju-liu-zhong-de-zhong-wei-shu-lcof/" target="_blank" rel="noopener noreferrer">剑指 Offer 41. 数据流中的中位数<svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg></a>『大顶堆+小顶堆』
<ul><li><a href="https://leetcode-cn.com/problems/find-median-from-data-stream/" target="_blank" rel="noopener noreferrer">295. 数据流的中位数<svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg></a>『重复题』</li></ul></li> <li><a href="https://leetcode-cn.com/problems/kth-largest-element-in-a-stream/" target="_blank" rel="noopener noreferrer">703. 数据流中的第 K 大元素<svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg></a>『大顶堆+小顶堆』
<ul><li><a href="https://leetcode-cn.com/problems/jBjn9C/" target="_blank" rel="noopener noreferrer">剑指 Offer II 059. 数据流的第 K 大数值<svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg></a>『重复题』</li></ul></li></ul> <h3 id="_3-1-1352-最后-k-个数的乘积"><a href="#_3-1-1352-最后-k-个数的乘积" class="header-anchor">#</a> 3.1.<a href="https://leetcode-cn.com/problems/product-of-the-last-k-numbers/" target="_blank" rel="noopener noreferrer">1352. 最后 K 个数的乘积<svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg></a></h3> <ul><li><a href="https://leetcode-cn.com/problems/product-of-the-last-k-numbers/" target="_blank" rel="noopener noreferrer">1352. 最后 K 个数的乘积<svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg></a></li> <li>坑点：可能有0</li> <li>『奇怪的技巧学习』+前缀乘</li></ul> <div class="language-cpp line-numbers-mode"><pre class="language-cpp"><code><span class="token keyword">class</span> <span class="token class-name">ProductOfNumbers</span> <span class="token punctuation">{</span>
<span class="token keyword">public</span><span class="token operator">:</span>
    <span class="token function">ProductOfNumbers</span><span class="token punctuation">(</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
        LastBlockLen<span class="token operator">=</span><span class="token number">0</span><span class="token punctuation">;</span>
        NoZeroVec<span class="token punctuation">.</span><span class="token function">resize</span><span class="token punctuation">(</span><span class="token number">40000</span><span class="token operator">+</span><span class="token number">5</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
        NoZeroVec<span class="token punctuation">[</span><span class="token number">0</span><span class="token punctuation">]</span><span class="token operator">=</span><span class="token number">1</span><span class="token punctuation">;</span>
    <span class="token punctuation">}</span>
    
    <span class="token keyword">void</span> <span class="token function">add</span><span class="token punctuation">(</span><span class="token keyword">int</span> num<span class="token punctuation">)</span> <span class="token punctuation">{</span>
        <span class="token keyword">if</span><span class="token punctuation">(</span> <span class="token number">0</span><span class="token operator">==</span>num <span class="token punctuation">)</span>
        <span class="token punctuation">{</span>
            LastBlockLen<span class="token operator">=</span><span class="token number">0</span><span class="token punctuation">;</span>
        <span class="token punctuation">}</span>
        <span class="token keyword">else</span>
        <span class="token punctuation">{</span>
            NoZeroVec<span class="token punctuation">[</span> <span class="token operator">++</span>LastBlockLen <span class="token punctuation">]</span><span class="token operator">=</span>num<span class="token punctuation">;</span>
            NoZeroVec<span class="token punctuation">[</span> LastBlockLen <span class="token punctuation">]</span><span class="token operator">*=</span>NoZeroVec<span class="token punctuation">[</span> LastBlockLen<span class="token operator">-</span><span class="token number">1</span> <span class="token punctuation">]</span><span class="token punctuation">;</span>
        <span class="token punctuation">}</span>
    <span class="token punctuation">}</span>
    
    <span class="token keyword">int</span> <span class="token function">getProduct</span><span class="token punctuation">(</span><span class="token keyword">int</span> k<span class="token punctuation">)</span> <span class="token punctuation">{</span>
        <span class="token keyword">if</span><span class="token punctuation">(</span> k<span class="token operator">&gt;</span>LastBlockLen <span class="token punctuation">)</span>
        <span class="token punctuation">{</span>
            <span class="token keyword">return</span> <span class="token number">0</span><span class="token punctuation">;</span>
        <span class="token punctuation">}</span>
        <span class="token keyword">else</span>
        <span class="token punctuation">{</span>
            <span class="token comment">//注意，也包含了特判『LastBlockLen==k』</span>
            <span class="token keyword">return</span> NoZeroVec<span class="token punctuation">[</span> LastBlockLen <span class="token punctuation">]</span><span class="token operator">/</span> NoZeroVec<span class="token punctuation">[</span> LastBlockLen<span class="token operator">-</span>k <span class="token punctuation">]</span><span class="token punctuation">;</span>
        <span class="token punctuation">}</span>
    <span class="token punctuation">}</span>
<span class="token keyword">private</span><span class="token operator">:</span>
    <span class="token keyword">int</span> LastBlockLen<span class="token punctuation">;</span>
    vector<span class="token operator">&lt;</span><span class="token keyword">int</span><span class="token operator">&gt;</span> NoZeroVec<span class="token punctuation">;</span>
<span class="token punctuation">}</span><span class="token punctuation">;</span>

<span class="token comment">/**
 * Your ProductOfNumbers object will be instantiated and called as such:
 * ProductOfNumbers* obj = new ProductOfNumbers();
 * obj-&gt;add(num);
 * int param_2 = obj-&gt;getProduct(k);
 */</span>
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